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WindowTitle->Lecture 11 MIT 3.016 (Fall 2011)   \251 W. Craig Carter  \
2003--2011
WindowFrame->Normal*)

(* Beginning of Notebook Content *)
Notebook[{
Cell["Visualizing Time-Dependent Scalar and Vector Fields", "Title"],

Cell[CellGroupData[{

Cell["Cross Products", "Subtitle"],

Cell["Here is the built-in cross product between two vectors", "Text"],

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Cell["\<\
And, here is the standard visual way to do it by hand with the determinant.\
\>", "Text"],

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Cell["Pick out each of the compenents to create the vector", "Text"],

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Cell["\<\
Check for equality between the old-fashioned way and Mathematica's built-in \
function\
\>", "Text"],

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Cell[CellGroupData[{

Cell["Visualizing Space Curves as Time-Dependent Vectors", "Subtitle",
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Cell["Create a trajectory of a point or a particle", "Text"],

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ParametricPlot3D allows us to visualize the entire trajectory at once.\
\>", "Text"],

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Cell["\<\
Here is a function to create a graphic with a variable end - point. We will \
have the function remember when it has already computed a graphic, trading \
memory for a possible speed-up.\
\>", "Text",
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Use manipulate on the graphics function to visualize how the curve develops \
with its parameter\
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However, we need to fix the length scale between frames, so we use the last \
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Next, we add a graphic element to show the vector, drawn from the origin, for \
each end-point.\
\>", "Text",
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Cell["Visualizing Local Tangent Vectors of Space Curves", "Subtitle",
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Cell[TextData[{
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Cell["Write out a function for the derivative:", "Text",
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Plot it.... Note that it doesn't \"appear\" to be periodic, which would be \
wrong.\
\>", "Text",
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The puzzle can be visualizing the time-development of both curves:\
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To visualize the \"tangency property,\"  we translate the derivative-vector \
to the end of the space curve\
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Cell["\<\
The Time-Dependent Solution to the Diffusion Equation in the Plane \
Rectangular Initial Conditions\
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Cell["\<\
This is the solution to the time-dependent diffusion equation in the infinite \
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\>", "Text",
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Visualize the solution by plotting the concentration as  function of x and y, \
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Visualizing Diffusional Flux: The Time-Dependent Gradient Field\
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